Context and antecedents

The ideal-gas theory begins with Bose’s counting of light quanta (Bose 1924) and Einstein’s extension to a monatomic ideal gas, including the condensation prediction in his second paper (Einstein 1924, 1925). Bogoliubov’s treatment of weakly interacting bosons introduced an approximate quasiparticle description based on macroscopic occupation of one mode (Bogolubov 1947).

For interacting particles, Penrose and Onsager formulated condensation through a macroscopic eigenvalue of the one-particle density matrix and applied this viewpoint to liquid helium (Penrose and Onsager 1956). Yang developed the broader framework of off-diagonal long-range order, covering Bose condensation and fermion pairing (Yang 1962). These criteria identify the order to be proved; establishing it for a specified interacting Hamiltonian remains a separate mathematical problem.

Rigorous lattice antecedents include Dyson, Lieb, and Simon’s low-temperature long-range order for the spin-\(\tfrac12\) nearest-neighbor quantum XY model on the simple cubic lattice in dimensions \(d\ge3\) (Dyson et al. 1978), and Kennedy, Lieb, and Shastry’s ground-state long-range order for all spins on hypercubic lattices in dimensions \(d\ge2\) (Kennedy et al. 1988). For spin-\(\tfrac12\), the spin–hard-core-boson correspondence relates these results to condensation of half-filled lattice bosons. Passing to a continuum Hamiltonian with a fixed interaction and a prescribed density requires additional arguments.

The distinction between the thermodynamic and Gross–Pitaevskii limits is essential. Lieb and Seiringer proved complete condensation in the Gross–Pitaevskii limit, with the interaction scaled so that \(Na\) remains fixed, where \(a\) is the scattering length (Lieb and Seiringer 2002). Deuchert and Seiringer subsequently treated positive-temperature homogeneous gases with scattering length \(a_N=a_vL/N\), where \(a_v\) is the scattering length of the unscaled potential, and controlled the one-particle density matrix of approximate Gibbs minimizers (Deuchert and Seiringer 2020). These results allow the interaction scale to vary with particle number. The problem considered here instead fixes the potential and a positive density before taking the volume to infinity.

The limits between Gross–Pitaevskii scaling and a fixed-density thermodynamic limit have several forms. Brennecke, Brooks, Caraci, and Oldenburg proved complete ground-state condensation in the Gross–Pitaevskii regime and beyond (Brennecke et al. 2025). Fournais treated low-energy states on growing boxes in a joint dilute and increasing-volume limit (Fournais 2021, Theorem 1.2). Chong, Liang, and Nam gave simpler proofs of low-energy condensation using kinetic localization and Poincaré-type inequalities (Chong et al. 2026). For the unit-box interaction \(N^{2-2\kappa}v(N^{1-\kappa}x)\), rescaling to a fixed potential gives density \(N^{3\kappa-2}\); the regimes \(\kappa<2/3\) therefore couple dilution to increasing particle number. Junge’s low-temperature canonical Gibbs-state result uses Neumann boxes whose size remains coupled to the dilute limit (Junge 2026, Corollary 6).

The thermodynamic energy problem has a separate history. Dyson’s hard-sphere bounds and the leading dilute asymptotics of Lieb and Yngvason established the first-order energy scale (Dyson 1957; Lieb and Yngvason 1998). The next correction was predicted by Lee, Huang, and Yang’s pseudopotential calculation (Lee et al. 1957). Fournais and Solovej proved the corresponding lower bound for nonnegative radial finite-range potentials, including hard cores (Fournais and Solovej 2023, Theorems 1.1 and 1.3); recent upper bounds attain the same coefficient for hard spheres and for general nonnegative even finite-range potentials (Basti et al. 2026b, 2026a). These energy results, and low-temperature free-energy estimates for strong potentials (Fournais et al. 2026), do not by themselves determine occupation of a single orbital: the one-particle kinetic gap tends to zero as the volume grows.

A distinct thermodynamic-limit result appears in Sütő’s two-step preprint (Sütő 2023). Its Theorem 1.2 assumes nonnegativity both of the potential and of its Fourier transform, together with integrability, a Fourier-moment condition, and spatial decay, and states condensation above a temperature-dependent density threshold. Fourier positivity is an additional restriction: for example, a nonnegative spherical square barrier has a Fourier transform that changes sign. That result therefore does not cover all bounded nonnegative radial finite-range potentials, and its fixed-temperature threshold does not alone provide the joint low-density and low-temperature uniformity sought here.

Galanda and Pinamonti construct algebraic equilibrium states with a condensate background under quantitative convergence conditions for a spatially localized interaction (Galanda and Pinamonti 2025, Theorem 6.8). Their subsequent analysis treats renormalized correlation functions in Gross–Pitaevskii scaling, with a zero-temperature limit in its final convergence theorem (Galanda and Pinamonti 2026, Theorem 7.3). These constructions use a different state formulation and coupled interaction, regulator, and temperature limits from the fixed-potential canonical thermodynamic limit considered here.

The trajectory formulation descends from Feynman’s treatment of the Bose gas (Feynman 1953); for the canonical Feynman–Kac representation and its symmetrization and one-particle normalizations, see (Ueltschi 2006, Appendix A). The random-path comparison has a different ancestry. Benjamini, Pemantle, and Peres constructed paths with exponentially decaying intersection tails (Benjamini et al. 1998). Häggström and Mossel developed a hierarchical random-drift construction with low predictability; its independent block variables are a close antecedent of the dyadic velocities used below (Häggström and Mossel 1998, sec. 3, Proposition 3.1). Abbe, Massoulié, Montanari, Sly, and Srivastava used finite-endpoint path averages and overlap estimates for group synchronization (Abbe et al. 2018); Garban and Spencer used this approach for classical disordered-spin symmetry breaking along the Nishimori line (Garban and Spencer 2023). Here the finite cell paths, their thickened intersection estimates, and the continuum bridge comparison require separate proofs.

How the comparison works

Let \(q_0=\langle u_0,\gamma^{(1)}u_0\rangle/N\). The Feynman–Kac formula expresses the partition function by positive weights on closed collections of paths. Replacing one matching endpoint by an independently integrated spatial point gives a positive measure \(\mu\) of total mass \(q_0\). We construct a measure \(\nu\) by changing a short portion of closed paths, with mass at least a fixed \(a>0\), and show that its density \(f=\,\mathrm d\nu/\,\mathrm d\mu\) has bounded second moment. Cauchy–Schwarz then gives \[a^2\le q_0\int f^2\,\mathrm d\mu.\] Thus the main problem is a comparison of positive path measures.

Partition space into cells of side \(s\) comparable to \(\rho^{-1/2}\), and change paths only during a time interval of length \(\delta=s^2\). The mean count \(m=Ns^3/V\) is comparable to \(\rho^{-1/2}\), making many particle choices available in a typical cell as the density decreases, while the slab remains short compared with \(1/T\) throughout the positive-temperature range.

The local analysis must control simultaneous failures in specified families of cells. Energy and entropy estimates first give occupation tails and separation of most endpoints at the shorter scale \(r=\zeta\rho^{-1/3}\), with \(\zeta>0\) fixed and small. A particle is eligible when its endpoints are separated, its path stays near them during short initial and final time intervals, and it satisfies overall confinement and integrated interaction-score bounds for the path ensemble. The interacting bridges are not independent. A separate localized partition-function comparison bounds their joint trajectory failures, with a cost proportional to the volume of the tested cells. Together with the one-time estimates and score bounds, this makes specified sets of unusable cells unlikely.

Between two distant cells, choose a random lattice path, called a skeleton, whose encounters with an independent copy have bounded exponential moments. Here an encounter records a vertex within a fixed cell distance of the other path; a second estimate weights all vertices by exponentially decaying distance. These two geometric bounds are established independently of the gas in Section 8.1. Detour around unusable components. Select one eligible particle from each cell on the resulting simple path, and cyclically reassign their starting points. One end is opened and the selected bridges are sampled from their exactly normalized interacting law. This defines \(\nu\).

The second moment involves two independently chosen paths in related configurations. Estimating both changes against the original closed configuration would incur a cost proportional to their lengths. Instead, a reverse insertion supplies a baseline in which contributions far from encounters cancel. Two effects remain small as \(\rho\) decreases: an encounter of distinct separated bridges has interaction cost tending to zero at rate \(r^{-1/2}\), and coincident particle choices have probability of order \(1/m\).

Changing trajectories also changes which cells are usable and which particles are eligible, so the two routes cannot be treated as independent of the gas. A stricter cell test in the reverse configuration controls these changes: away from encounters the selection weights agree, while rare components give a controlled list of possible detours. Keeping those selection weights throughout the comparison yields the required second-moment bound after averaging over the environment and the two paths.

Sections [sec:energy]–7 prove the local analytic bounds. Section 8 constructs the paths and detours. Section 9 defines the change of measure, and Section 10 proves its second-moment estimate with all normalizations and selection constraints retained. Section 11 fixes the parameters in order and passes the uniform positive-temperature bound to zero temperature.

A localized kernel for one marked bridge

Partition the failure event into disjoint bins. In bin \(j\ge0\), the lifted maximum distance from the end belongs to \((D_js,2D_js]\). Bin \(*\) consists of a failure of one of the two short-interval tests with no macro failure. Write \[h_j=D_j,\quad b_j=D_j^2\quad(j\ge0), \qquad h_*=D,\quad b_*=r.\] Here and below a sum over bins includes \(*\). Let \[\Omega_j=\{x\in\Lambda_L:d(x,U)<(2h_j+4)s\}.\] If this set is the whole torus, its Laplacian has periodic boundary conditions; otherwise we use the Dirichlet Laplacian. Choose a Lipschitz function \(0\le p_j\le1\), equal to one when \(d(x,U)\le2h_js\), vanishing when \(d(x,U)\ge(2h_j+2)s\), and satisfying \(|\nabla p_j|\le C/s\). It has support compactly contained in \(\Omega_j\) unless \(\Omega_j\) is the whole torus. The elementary coarse-cell cover bound gives \[|\Omega_j|\le C h_j^3|U|, \qquad \Omega_j\subset B_j,\qquad |B_j|\le C h_j^3|U|,\] for suitable coarse-cell unions \(B_j\). The constants also work when the enlarged set covers or winds around the torus.

Let \(K_j(E,S)\) be the unnormalized free single-bridge kernel with end \(E\in U\) and failure in bin \(j\). There are absolute constants \(c,C>0\) and a fixed \(c_1>1\) such that \[ K_j(E,S) \le C e^{-cb_j}p_j(E)p_j(S) [e^{c_1\delta\Delta_{\Omega_j}}](E,S). \tag{18} \] These constants are independent of \(D\) and \(\zeta\) after the small-density restrictions below are imposed.

To prove this, fix a lift of \(E\) and consider separately each lift of \(S\), with displacement \(d\) from that lift of \(E\). Put \(k_t(d)=(4\pi t)^{-3/2}e^{-|d|^2/(4t)}\). In a macro bin the event forces \(|d|\le2h_js\). If \(|d|>h_js/2\), widening the Gaussian gives \[k_\delta(d)\le C e^{-ch_j^2}k_{c_1\delta}(d).\] If \(|d|\le h_js/2\), the zero-end bridge fluctuation must have maximum at least \(h_js/2\). The Brownian bridge maximum tail is at most \(Ce^{-ch_j^2}\), while \(k_\delta\le c_1^{3/2}k_{c_1\delta}\). This proves the same bound for this case.

For bin \(*\) one has \(|d|\le Ds\). Once \(s\ge40D\), the linear drift on a time interval of length \(r\) is at most \(r/40\). If \(B\) is Brownian motion with generator \(\Delta\), the fluctuation is \(B_t-(t/\delta)B_\delta\). The maximal inequality on \([0,r]\) bounds the chance that the first term exceeds a fixed positive multiple of \(r\) by \(Ce^{-cr}\). The corresponding estimate for the second term is \(Ce^{-c\delta}\), since exceeding that multiple of \(r\) requires \(|B_\delta|\ge c\delta\). We take \(r\le\delta/2\). Time reversal gives the same bound at the other endpoint. Thus the bin-\(*\) kernel for each retained lift is at most \(Ce^{-cr}k_{c_1\delta}(d)\).

It remains to justify the Dirichlet kernel on the right of [eq:18], rather than a free kernel without localization. For every retained lift, the straight segment from \(E\) to \(S\) is within distance \(2h_js\) of the chosen lift of \(E\). Every lifted path at distance less than \(s\) from that segment projects into \(\Omega_j\). A bridge of diffusion coefficient \(c_1\) over time \(\delta=s^2\) has probability at least a fixed \(q_*>0\) of remaining in this tube, by scaling its zero-end fluctuation. Consequently the killed torus kernel is at least \[q_*\sum_{\substack{\text{retained lifts of }S}} k_{c_1\delta}(d).\] The terms are disjoint winding classes of torus paths. This remains valid for extents greater than \(L\) and for \(\Omega_j=\Lambda_L\). On the support of \(K_j\), both \(p_j(E)\) and \(p_j(S)\) equal one. Summing the preceding bounds over lifts proves [eq:18].

The colored operator and its large powers

Choose \(\kappa>0\) sufficiently small, and set \[\eta_j=e^{-\kappa b_j},\qquad M_j=\kappa b_j/\delta.\] By increasing the lower restrictions on \(D\) and \(r\), the prefactor in [eq:18], including the marking factor \(e-1\), is bounded by \(\eta_j e^{-\delta M_j}=e^{-2\kappa b_j}\). The choice of \(\kappa\) is independent of \(D\) and \(\zeta\).

We will represent marked bridges by auxiliary particle colors. The small coefficient \(\eta_j\) limits the enlargement of the map into the colored space, while the positive gap \(M_j\) will pay the free-energy cost of removing a particle from the original interacting color. Normalizing this map will introduce a local attractive potential. The density estimate controls its cost on the original color, and the gap also absorbs it on the marked colors.

The enlarged one-particle Hilbert space is \[\mathfrak h_e=L^2(\Lambda_L) \oplus\bigoplus_j L^2(\Omega_j),\] with the first summand called the original color and the others indexed by the failure bins. Define \[v_e f=\bigl(f,(\sqrt{\eta_j}p_jf)_j\bigr), \qquad V_e=v_e^{\otimes N}.\] The restrictions to the relevant domains in each marked component are understood. On the labeled \(N\)-fold tensor product, define \(\mathbb H\) by its closed quadratic form, separately in each color assignment. Original-color particles have their original kinetic energy and the full pair potential between themselves. A particle of color \(j\) has one-body operator \(-c_1\Delta_{\Omega_j}+M_j\). All interactions incident to marked particles are omitted. The operator and the embedding commute with, or intertwine, coordinate permutations. They therefore restrict to the symmetric \(N\)-particle spaces. On those spaces put \[\mathcal T=V_e^*e^{-\delta\mathbb H}V_e.\]

Indeed, expanding \[e^{J_U^{\mathrm{fail}}} =\prod_{i=1}^N\left(1+(e-1) \mathbf1_{\{E_i\in U,\ i\text{ fails}\}}\right)\] marks subsets of labels. For each marked subset, decompose its failure events into bins and drop precisely the interactions incident to marked labels. This increases the nonnegative path weight. Applying [eq:18] to each marked path leaves the original interacting kernel on the other labels. The sum over all color assignments is exactly the kernel of \(\mathcal T\). Thus \(\mathcal T\) majorizes the tested slab kernel pointwise on labeled coordinates. The nonnegative outer kernel, including its symmetrization projector, gives \[\mathbb E_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}} \le Z_N^{-1}\mathop{\mathrm{Tr}}_{\mathrm{sym}} e^{-(\beta-\delta)H}\mathcal T.\] For \(\beta>\delta\), Schatten Hölder with conjugate exponents \(\beta/(\beta-\delta)\) and \(\beta/\delta\) yields \[ \mathbb E_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}} \le \left(\frac{\mathop{\mathrm{Tr}}_{\mathrm{sym}}\mathcal T^{\beta/\delta}} {Z_N}\right)^{\delta/\beta}. \tag{19} \]

The embedding has norm larger than one. We next account for its effect on the high power in [eq:19] without losing the spatial localization. Define the bounded Lipschitz function \[a(x)=\frac12\log\left(1+\sum_j\eta_jp_j(x)^2\right), \qquad A=\sum_{i=1}^N a(x_i).\] The same position multiplier acts in every color. One has \(V_e=e^A U_e\), where \(U_e\) is an isometry: at the one-particle level the squared norm of the color vector is \(e^{2a(x)}\). Consequently \[\mathcal T=U_e^*XU_e,\qquad X=e^Ae^{-\delta\mathbb H}e^A.\] Eigenvalue min–max for a positive compact compression gives \(\lambda_k(\mathcal T)\le\lambda_k(X)\) for every \(k\), including multiplicities. Hence the trace of any positive power of \(\mathcal T\) is bounded by that of \(X\).

The partition-function estimate with confined colors

The preceding lemma has converted the large power of the marked-slab operator into an ordinary heat trace. It remains to compare that trace with the original partition function. Spatial confinement is essential here: it makes the final cost proportional to \(|U|\), the volume where failures are tested, rather than to the whole torus.

Set \[\Sigma=\sum_j\eta_jh_j^3 =\sum_{j\ge0}D_j^3e^{-\kappa D_j^2} +D^3e^{-\kappa r}.\] From the definition of \(a\) and Cauchy–Schwarz, \[0\le W_0\le\frac C\delta\sum_j\eta_j\mathbf1_{\Omega_j}.\] For example, \[|\nabla a|^2 =\frac{|\sum_j\eta_jp_j\nabla p_j|^2} {(1+\sum_j\eta_jp_j^2)^2} \le\frac C\delta\sum_j\eta_j\mathbf1_{\Omega_j}.\] All constants are independent of the number or sizes of the domains. Taking \(D\) large and then \(\rho\) small makes \(\|W_0\|_\infty\le\rho\), because \(\delta\ge K^2\rho^{-1}\) and \(r\to\infty\).

For \(0\le J\le N\), differentiate the original-color partition function in the strength of \(-W_0\). The density input, applied to the covers \(B_j\) and to every external potential \(-tW_0\) with \(0\le t\le1\), gives \[\log Z_J^{(-W_0)}-\log Z_J =\beta\int_0^1\langle n_{W_0}\rangle_{J,-tW_0}\,\,\mathrm dt \le\frac{\beta C_v\rho|U|}\delta\Sigma.\] The free-energy increment input also gives \(Z_J/Z_N\le e^{\beta C_v\rho(N-J)}\).

To use these inequalities, decompose the enlarged symmetric space according to the occupation of each color: \[\operatorname{Sym}^N\mathfrak h_e \cong \bigoplus_{J+\sum_j k_j=N} \operatorname{Sym}^J L^2(\Lambda_L) \otimes\bigotimes_j\operatorname{Sym}^{k_j}L^2(\Omega_j).\] Each number sector is one tensor product. There is no multinomial multiplicity in its trace: the normalized symmetrization over color assignments is the unitary realizing this decomposition. In that sector, the cost \(e^{\beta C_v\rho(N-J)}\) is one factor per marked particle. On color \(j\), subtracting \(W_0\) and this cost leaves an operator bounded below by \[-c_1\Delta_{\Omega_j} +M_j-\|W_0\|_\infty-C_v\rho \ge -c_1\Delta_{\Omega_j}+M_j/2.\] For this last inequality use \(M_j\ge\kappa b_j\rho/(4K^2)\), choose \(D\) sufficiently large depending on \(v,K\), and then make \(r\) sufficiently large. The same choices ensure \(\beta M_j/2\ge1\) in every color.

Sum the marked occupation numbers and drop their total-number constraint. This bounds the sum by a product of ideal Bose grand partition functions with gaps \(M_j/2\). In each color its logarithm is at most \[\sum_{\ell\ge1}\frac{e^{-\ell\beta M_j/2}}\ell \mathop{\mathrm{Tr}}e^{c_1\ell\beta\Delta_{\Omega_j}} \le C|\Omega_j|(V^{-1}+T^{3/2})e^{-\beta M_j/2}.\] The estimate uses the killed heat-kernel bound by the periodic one and the geometric series with \(\beta M_j/2\ge1\). The resulting sum over colors is finite, so the passage from finitely many colors to countably many is justified by monotone convergence of the nonnegative occupation sums. Combining this bound with [eq:19], the compression inequality, and [eq:20], we obtain the quantitative estimate \[\begin{equation*} \log\mathbb E_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}} \le C_v\rho|U|\Sigma +C\delta T|U|(V^{-1}+T^{3/2}) \sum_jh_j^3e^{-\beta M_j/2}. \end{equation*}\]

This also proves the required uniformity. Since \(\beta\ge\delta\), the color sum is bounded at fixed \(D\) for all sufficiently small \(\rho\), using the summable macro terms \(D_j^3e^{-\kappa D_j^2/2}\) and the micro term \(D^3e^{-\kappa r/2}\). Moreover \(V\ge s^3\), \(\delta\le4K^2/\rho\), and \(T\le\rho^2\), so \[\frac{\delta T}{\rho}(V^{-1}+T^{3/2}) \le C_K(\rho^{3/2}+\rho^3).\] The micro contribution \(D^3e^{-\kappa r}\) to \(\Sigma\) tends to zero at fixed \(D,\zeta\). Substituting these estimates in the partition-function bound proves [eq:17], with all errors uniform in the temperature and the thermodynamic volume, as asserted.

Random skeletons and their encounter moments

The use of unpredictable paths and intersection moments follows Benjamini–Pemantle–Peres (Benjamini et al. 1998, Theorem 1.3 and Lemma 3.1). The independent block velocities below are a continuous dyadic version of the aligned block-drift construction of Häggström and Mossel (Häggström and Mossel 1998, sec. 3, Proposition 3.1). Finite-endpoint path averaging appears in the synchronization argument of Abbe, Massoulié, Montanari, Sly, and Srivastava (Abbe et al. 2018, sec. 6, Lemma 6.1) and in the symmetry-breaking argument of Garban and Spencer (Garban and Spencer 2023, sec. 2, Lemma 2.5). We give an explicit dyadic construction and prove the thickened and soft-distance estimates needed here.

The construction in this subsection is purely geometric. Let \(\mathcal G_n=(\mathbb Z/n\mathbb Z)^3\), and fix an ordered pair \(a,b\) with lifted displacement \((w,d_2,d_3)\) satisfying \[\lceil n/10\rceil\le w\le\lfloor n/8\rfloor, \qquad |d_2|,|d_3|\le w.\] Call such a pair admissible. Admissible pairs form a fraction at least an absolute \(\upsilon>0\) of all ordered pairs for sufficiently large \(n\). All coordinates in the construction are first taken relative to a fixed lift of \(a\), and then projected to \(\mathcal G_n\).

The first coordinate will advance from \(0\) to \(w\). In the other two coordinates we add a random displacement to the straight interpolation between the endpoints. We read that displacement forward up to the midpoint and backward afterward, so it vanishes at both endpoints. This fixes the endpoint without conditioning the random increments; the two halves deliberately use the same displacement variables.

For \(q=2,3\), let \((U^{(q)}_{\ell,j})_{\ell,j\ge0}\) be mutually independent random variables, uniform on \([-1,1]\). Fix \(c_{\rm vel}=(1-2^{-1/4})/2\), and set \[v_i^{(q)}=c_{\rm vel}\sum_{\ell\ge0}2^{-\ell/4} U^{(q)}_{\ell,\lfloor(i-1)/2^\ell\rfloor}, \qquad Z_t^{(q)}=\sum_{i=1}^t v_i^{(q)},\qquad Z_0^{(q)}=0.\] The series converges absolutely, and \(|v_i^{(q)}|\le1/2\). At integer times \(t=0,\ldots,w\) use the base vertices \[\left(t,\left\lfloor td_2/w+Z^{(2)}_{\min(t,w-t)}\right\rfloor, \left\lfloor td_3/w+Z^{(3)}_{\min(t,w-t)}\right\rfloor\right).\] Starting from the base vertex at time \(t\), first advance one step in the first coordinate, then take the signed second-coordinate steps, and finally the signed third-coordinate steps needed to reach the base vertex at time \(t+1\). Omit zero-length moves. Begin the vertex list with \(a\) and append each newly reached vertex once, without repeating a shared endpoint of successive pieces. Each transverse coordinate changes by at most two; thus the resulting nearest-neighbor skeleton \(\gamma=(z_0,\ldots,z_l)\) has at most five steps per time interval and at most five listed occurrences at each first-coordinate level \(t\ge1\), with one occurrence at level zero. The law depends only on the torus and the ordered endpoints. Independent skeletons use independent arrays of the \(U\)’s. The dyadic weights are chosen so that a previously unused block of \(k\) increments supplies a transverse displacement of order \(k^{3/4}\). In two transverse coordinates this gives the summable conditional density bound \(Ck^{-3/2}\) proved below.

For vertices on the torus write \[d_\infty(x,y)=\min_{k\in\mathbb Z^3}\|\widetilde x-\widetilde y+nk\|_\infty, \qquad d_\infty(x,\gamma')=\min_{z'\in\gamma'}d_\infty(x,z'),\] where the tildes denote arbitrary lifts. For two independent skeletons \(\gamma,\gamma'\) with the same ordered endpoints and a fixed radius \(R_*\ge1\), define \[J_0=\sum_{z\in\gamma}\mathbf1_{\{d_\infty(z,\gamma')\le R_*\}} +\sum_{z'\in\gamma'}\mathbf1_{\{d_\infty(z',\gamma)\le R_*\}}, \qquad S_B=\sum_{z\in\gamma}e^{-d_\infty(z,\gamma')/B}\quad(B>0).\] Every sum uses the listed vertex occurrences, including the endpoints. An occurrence contributes once to \(J_0\) when at least one occurrence of the other skeleton lies within the stated distance.

Detours around unusable components

We now apply the geometric construction to the coarse grid of Section 2. This is the point at which the physical cutoffs enter: set \[R_* = \lceil20(R+R_A+D+20)\rceil,\qquad \varepsilon=\alpha/20,\] decreasing the absolute constant in \(\alpha\) if necessary. Choose \(D\) large enough for the earlier conditions with this choice. This is possible because [eq:17] has Gaussian decay in \(D\), whereas \(R_*\) grows only linearly. Choose \(\zeta,C_0\) and then the small-density threshold as in Proposition 13. Increase \(C_0\) also so that, for two uniform anchors, the probability of any end-score failure at threshold \(C_0\) is less than \(\upsilon/8\); the spatial average after [eq:16] gives this last requirement.

The skeleton law does not see the unusable cells. We next adapt each sampled skeleton to those cells by a deterministic detour procedure. This step uses nearest-neighbor components of bad cells but allows sup-norm-one steps on the resulting path. Here and below, bad means unusable in the sense of Section 5. Reject the configuration if any nearest-neighbor component of bad cells has size \(\ge n/4\).

A smaller component \(T\) lifts consistently to a finite connected lattice set, up to translation. Indeed, simple cycles in it are too short to wrap, and closed walks decompose into simple cycles and backtracks. Its bounding box has coordinate diameters less than \(n/4\), since two of its vertices can be joined within \(T\) by fewer than \(n/4\) steps. Define its fill by adjoining the finite nearest-neighbor components of its lattice complement, and project the result back to the torus. The fill lies in the same bounding box: a vertex outside that box connects straight outwards to infinity without meeting \(T\). Reject if either endpoint lies in any projected fill.

The detour procedure scans an accepted skeleton. On entering an outermost fill, meaning one not contained in another fill, it replaces the portion through its last visit there by an exterior-boundary path. The next proof establishes the boundary connectivity that permits this replacement with sup-norm-one steps. Fix all tie-breaking rules for boundary paths in advance, and use chronological loop erasure for the resulting walk.

Rejection by large components has closed probability tending to zero at large \(n\) by (22): count connected sets of size \(\lceil n/4\rceil\) by at most \(M C^{\lceil n/4\rceil}\) (walk a rooted spanning tree), and a larger component contains one. For either given endpoint, containment in a small fill has probability at most \(\sum_{j\ge1} C(j+1)^3 C^j q_{\rm cl}^j\), since its bad component of size \(j\) must be within distance \(j\). This can be made arbitrarily small by the density restriction.

The base law and acceptance

Our base law is the closed probability [eq:14], denoted by the configuration \(C\), together with independent uniform points \(x,z\) of law \(dx\,dz/V^2\). In the slab notation, configurations are tuples of paths rather than density operators; their starting tuples carry superscripts, such as \(S^C\), and a subscript restricts their paths to a set of slots.

Set \(a=\operatorname{cell}(x)\) and \(b=\operatorname{cell}(z)\). The first index of \(\mu_{ab}\) in Section 7 specifies the new inner endpoint’s cell, and the second specifies the outer endpoint’s cell. Thus the two orientations used below are \[\begin{array}{c|c|c|c} &\text{opened slot}&\text{outer endpoint}&\text{new inner endpoint}\\ \hline \text{forward insertion }\mu_{ba}&i_0&Y_{i_0}\in a&z\in b\\ \text{reverse insertion }\mu_{ab}&e&Y_e\in b&x\in a. \end{array}\] The reverse slot \(e\) will be the last label on the selected path. Only valid ordered pairs, meaning the admissible pairs of Section 8.1, are used. For such a pair draw a skeleton and apply the deterministic adaptation to the unusable cells of \(C\). Require geometric acceptance and the two tests \(\mathcal A^+(x;C),\mathcal A^+(z;C)\le C_0\). Denote these combined conditions by \(\operatorname{acc}(C,\gamma)\). For an invalid pair the acceptance indicator and all weights below are zero. The base average acceptance mass is at least \(\upsilon/2\): the valid-pair fraction is at least \(\upsilon\), and the geometric rejection and dummy-score probabilities can each be made smaller than a fixed fraction of it by the choices already specified.

Selecting one eligible slot in each path cell

On acceptance let \(P(C,\gamma)\) be the ordered simple star path. For a cell \(Q\), write \[g_Q(C)=\sum_{i:Y_i\in Q}\mathbf1_{\{i\text{ eligible in }C\}}.\] Independently in each path cell choose a uniform eligible slot. This gives an ordered tuple \(I=(i_0,\ldots,i_k)\), one label per cell. For any candidate tuple define, on acceptance, \[\begin{split} h_\gamma(C,I)={}& \mathbf1_{\{(\operatorname{cell}(Y_{i_0}),\ldots, \operatorname{cell}(Y_{i_k}))=P(C,\gamma)\}}\\ &\quad\cdot\prod_{j=0}^k \frac{\mathbf1_{\{i_j\text{ eligible in }C\}}} {g_{\operatorname{cell}(Y_{i_j})}(C)}. \end{split}\] Set the weight to zero off acceptance, and interpret a factor with zero eligible count as zero. The cells of \(P\) are distinct and usable, so on acceptance the displayed factors are defined and \(\sum_Ih_\gamma(C,I)=1\). Equivalently, the sum ranges over candidate simple star paths from \(a\) to \(b\) and label choices in their outer \(Y\)-cells. This range depends on \(a,b,Y\), whereas the weight retains all dependence on the actual paths of \(C\).

The conditional interacting proposal

For a tuple \(I\), use cyclic predecessor and successor, acting as the identity off the tuple, and let \((p_I S)_i=S_{\operatorname{pred}_I(i)}\). Transform the starts to \(S^O=p_I S^C\). The inner ends \(E^O\) agree with \(Y\) except at slot \(i_0\), whose end is replaced by \(z\); paths off \(I\) are kept. Resample paths in \(I\) from their conditional interacting distribution restricted by \(\mathrm{cut}_R\).

More precisely, for a state \(X\) with fixed endpoints and complement paths, let \(W_G^X\) use the free bridge measures on \(G\) and the exponential of minus the interactions incident to \(G\), each pair once. Write \(\mathcal Z_G(X)=\int\mathbf 1_{\mathrm{cut}_R(G)} W_G^X\). The proposal on \(I\) uses this cut measure in \(O\), divided by \(\mathcal Z_I(O)\). The new displacements are at most \((D+10)s\) by eligibility and star adjacency: the start from the last cell goes to the end \(z\). Thus the normalization is positive.

The density and its second moment

Averaging a path-dependent likelihood before estimating its second moment also appears in random-trail detection and perturbed-lattice reassignment (Arias-Castro et al. 2008; Berger and Peres 2013; Peres and Sly 2014). In the present setting the conditional interacting measures and both proposal normalizers must be retained explicitly.

Write \(\nu\) for the resulting subprobability measure on the open variables. Its mass is the base acceptance probability, hence at least \(\upsilon/2\). Its density relative to \(\mu\) is \[f = m\,\mathbb E_\gamma\sum_{I:\,\mathrm{first}(I)=i_0} \frac{\mathbf 1_{\mathrm{cut}_R(O_I)}}{\mathcal Z_I(O)} \int W_I^{\widehat C}\, h_\gamma(\widehat C,I), \tag{26}\] where the hypothetical closed state \(\widehat C\) has \(S^{\widehat C}=p_I^{-1}S^O\), \(E^{\widehat C}=Y\), and paths off \(I\) as in \(O\). A term with zero old weight is set to zero even if its normalizer vanishes; positive old weight gives the displacement condition and a positive normalizer.

To verify the density, change start variables for each candidate tuple. The Jacobian is one and the outer kernel is invariant under this permutation. Write \(W_{\bar I}^{\rm int}\) for the complement bridge measures with only their internal interactions. This factor agrees before and after the change. Integrating the old paths on \(I\) leaves \[W_{\bar I}^{\rm int} \left[\int W_I^{\widehat C} h_\gamma(\widehat C,I)\right] \mathbf 1_{\mathrm{cut}_R(O_I)}W_I^O/\mathcal Z_I(O)\] (all incident weights and normalizers use the displayed fixed complement). The corresponding factor in the open measure is \(W_{\bar I}^{\rm int}W_I^O\). The remaining normalization comes from the label and dummy coordinate: \[\frac{\text{base density of }x} {\text{target index factor}\times\text{target density of }x} =\frac{V^{-1}}{N^{-1}s^{-3}}=\frac{N}{M}=m.\] Here both measures restrict \(x\) to the cell containing \(Y_{i_0}\), and both use \(dz/V\). This proves [eq:26], equivalently by integration against nonnegative test functions. Skeleton expectations are for the given cell pair, with zero weight for invalid pairs.

Cauchy–Schwarz gives \((\upsilon/2)^2\le q_0\int f^2\,\mathrm d\mu\). To compute the second moment, use \(\int f^2\,\mathrm d\mu=\int f\,\mathrm d\nu\): evaluate [eq:26] at a transformation from the base with independent skeleton \(\gamma'\) and tuple \(I'\). The resulting integral is \[\begin{split} &m\,\mathbb E_{x,z,\gamma',\gamma}\, Z_N^{-1}\int F(Y,S^C)\,dY\,dS^C \sum_{I',I:\,\mathrm{first}(I')=\mathrm{first}(I)}\int W(C) h_{\gamma'}(C,I')\\ &\hspace{30mm}\cdot \frac{\mathbf 1_{\mathrm{cut}_R(O_{I'})}W_{I'}^O} {\mathcal Z_{I'}(O)} \frac{\mathbf 1_{\mathrm{cut}_R(O_I)}}{\mathcal Z_I(O)} W_I^{\widehat C} h_\gamma(\widehat C,I). \end{split} \tag{27}\] The path integrations are over all closed paths, then proposed paths on \(I'\), and finally hypothetical old paths on \(I\). Here \(W(C)\) is the full slab weight, \(S^O=p_{I'}S^C\), and \(x,z\) have their base uniform law. Both tuples must have the same first slot, which is summed over. All integrands are nonnegative, so Tonelli permits the changes in integration order used next. Section 10 bounds [eq:27] uniformly.

The reverse insertion and the slot partition

Fix a term of [eq:27]. The tuple \(I'\) is selected along \(P'=P(C,\gamma')\) and produces the open state \(O\); the density at \(O\) uses a tuple \(I\) along \(P=P(\widehat C,\gamma)\). Their first label \(i_0\) is common. Write \(e\) for the last label of \(I\), and introduce the reverse state \(B\) by the following endpoint assignments: \[\begin{array}{c|c|c} \text{state}&\text{starting tuple}&\text{inner ending tuple}\\ \hline C&S^C&Y\\ O&p_{I'}S^C&Y\text{ with }Y_{i_0}\text{ replaced by }z\\ \widehat C&p_I^{-1}p_{I'}S^C&Y\\ B&p_I^{-1}S^C&Y\text{ with }Y_e\text{ replaced by }x. \end{array}\] The outer tuple \(Y\) is unchanged in all four states. The table specifies endpoints; the path variables retained or integrated in each state are described next. Figure 1 shows the two single-tuple assignments. In the cross term, the forward state uses \(I'\) and the reverse state uses \(I\).

Identify a tuple with its slot set when taking intersections and differences, and partition the selected slots as \[\mathsf h=I\cap I',\quad \mathsf d=\mathsf h\cup\operatorname{pred}_I(\mathsf h),\quad \mathsf j=I\setminus\mathsf d,\quad \mathsf l=I'\setminus\mathsf h,\quad \mathsf L=(I\cup I')^c .\] Thus \(\mathsf h\) contains the shared labels, while \(\mathsf d\) also contains their predecessors on \(I\). In particular \(|\mathsf d|\le2|\mathsf h|\) and \(e\in\mathsf d\), because the first label is shared. The two unshared portions that can cancel are \(\mathsf j\) and \(\mathsf l\); \(\mathsf L\) contains the untouched slots. Let \(J\) count cells of \(P\) within distance \(R_*\) of \(P'\) and conversely, adding both counts. Call these occurrences near and all others far.

All \(\mathsf L\) paths agree throughout. On \(\mathsf j\), \(C=O\) for endpoints and paths since the slots are off \(I'\), while \(B\) has the endpoints of \(\widehat C\): the \(I\)-successor is not shared, hence also off \(I'\), and \(S_i^B=S_{\operatorname{succ}_I(i)}^C=S_{\operatorname{succ}_I(i)}^O=S_i^{\widehat C}\) there. On \(\mathsf l\), \(B\) has the endpoints of \(C\), and \(\widehat C=O\). The separate path integrations off \(\mathsf L\) and our identification for the comparison are thus: \[\begin{array}{c|l|l} \text{set} & \text{path variables in (27)} & \text{use in }B\\ \hline \mathsf j & C_{\mathsf j}=O_{\mathsf j},\ \widehat C_{\mathsf j} & B_{\mathsf j}=\widehat C_{\mathsf j}\ \text{initially}\\ \mathsf l & C_{\mathsf l},\ O_{\mathsf l}=\widehat C_{\mathsf l} & B_{\mathsf l}=C_{\mathsf l}\\ \mathsf d & C_{\mathsf d},\ \widehat C_{\mathsf d},\ O_{\mathsf h} & B_{\mathsf d}\ \text{fresh}. \end{array}\] We will also reintegrate some of \(\mathsf j\) below; \(O\) on \(\mathsf d\setminus\mathsf h\) equals \(C\) there.

We work first on the domain of the original nonzero weights and cuts in (27). All paths there on \(I\cup I'\) in \(C,O,\widehat C\) satisfy \(\mathrm{cut}_R\), by the two cuts in (27) and the two eligibilities. Displacements even for all endpoints needed in conditional integrations are at most \((D+10)s\): in \(O\) use the forward displacement rule from \(C\) on \(I'\) and from \(\widehat C\) on \(I\) (indeed \(S^O=p_I S^{\widehat C}\)). This also covers the non-selected slots on the union in \(C,\widehat C\), which individually have the same endpoints as in \(O\). For \(B\) on \(\mathsf d\), if the successor is shared use its start from an eligible slot of \(C\), close to the successor cell (for \(e\) use that \(x\) is in the first cell); otherwise the start agrees with \(\widehat C\).

For the beginning and end anchors of the integrated rows where (16) is needed the scores against \(\mathsf L\) are \(\le C_0\). In \(O\) the starts on \(I'\) come from eligible starts of \(C\), on \(I\) from eligible starts of \(\widehat C\); the respective end anchors are covered by eligibility or the dummy \(z\) test. Similarly \(B_{\mathsf j},B_{\mathsf l}\) use eligible endpoints from \(\widehat C,C\); on \(\mathsf d\) use the successor rule just given and end eligibility of \(\widehat C\) or the dummy \(x\) test. Paths in \(\mathsf L\) are the same in the scores. Also any pair between \(\mathsf j,\mathsf l\) in \(O\), and in \(B\), has starts separated by more than \(r\) and likewise ends: by the indicated eligible sources the starts used on both sets are isolated in the common start multiset, of which each of \(O,B\) assigns distinct elements (indices) to distinct slots. These ordinary end slots are isolated among \(Y\) by eligibility in \(\widehat C,C\) respectively (no separation from the exceptional inner end is needed).

All confined bridge measures use nearest lifts. Write \(\pi_i^X\) for the probability bridge on that lift before imposing the cut, and factor out the kernels \(k\) of those individual lifts (not the periodic sums), with product denoted \(K_X(G)\) on a set \(G\). Factor the kernels from the cut normalizations as well (write \(\overline{\mathcal Z}\) for the remaining normalization). The proposed-path kernels cancel, and relative to using baseline kernels the ratio left in (27) off \(\mathsf L\) is \[\frac{K_C(I\cup I') K_{\widehat C}(I)}{K_B(I\cup I') K_O(I)} =\frac{K_C(\mathsf d)K_{\widehat C}(\mathsf d)} {K_B(\mathsf d)K_O(\mathsf d)}\le e^{C|\mathsf h|}. \tag{28}\]

Conditional laws and the two denominator bounds

For \(G=\mathsf j\) or \(\mathsf l\) write \(\Phi_G^X\) for the exponential of minus the interactions internal to \(G\) and with \(\mathsf L\) only, and \(Z[G,X]\) for its integral with the probability bridges and cuts \(R\), fixing \(\mathsf L\). Use empty-product conventions.

Restoring the baseline interaction weight

We describe the weight restoration in \(B\). Put in \(\mathsf j_n\subseteq\mathsf j\) those with end cell within \(2R+5\) of some \(\mathsf l\) cell; other \(\mathsf j\) cannot interact with \(\mathsf l\) on the cuts.

Tracking the constraints in this comparison

The original choices depend on which cells are usable and which slots are eligible. We now compare those tests between any original witness and every replacement of \(B_{\mathsf j_n},B_{\mathsf d}\) on the cuts, with the other data fixed as above. This universal comparison will justify using [eq:restoring-weights] inside the constrained integral.

In \(B\), use the outer tuple \(Y\), including \(Y_e\), for cell counts and end isolation, and declare the exceptional slot \(e\) ineligible. Other eligibility tests use the starts and paths as before; the exceptional path is included in all score sums. Let \(S_0(B)\) be the resulting unusable cells. Define \(S_1(B)\) by the stricter score threshold \(C_0-\Delta_A\) and the stronger required eligible fraction \(1-\varepsilon/2\), with the same occupancy interval. Thus \(S_0(B)\subseteq S_1(B)\). Write \(S(C),S(\widehat C)\) for the unusable cells in the two witness states. The table compares the tests; its last column gives the ineligible fraction causing failure when the count lies in the allowed interval. \[\begin{array}{c|c|c|c} \text{bad-cell set}&\text{allowed count}&\text{score cutoff} &\text{failure fraction}\\ \hline S(C),S(\widehat C),S_0(B)&[b_-m,b_+m]&C_0&>\varepsilon\\ S_1(B)&[b_-m,b_+m]&C_0-\Delta_A&>\varepsilon/2\\ \text{defective cells} &[2b_-m,b_+m/2]&C_0-\Delta_A&\ge\varepsilon/4. \end{array}\] Counts outside the displayed interval also cause failure. The symmetric defective test uses actual inner endpoints, whereas \(S_0(B),S_1(B)\) use the outer-end convention just specified. The witness states \(C\) and \(\widehat C\) are closed, so their inner ends are \(Y\).

Summing the comparison

The conditional comparison is now complete, including its selection predicates. We sum first over labels at fixed baseline paths, then over the finite lists of possible cell paths. The remaining environment average will be controlled by the insertion defect bound and the two skeleton moments.